Suppose you want to stand on one leg. Doing it eyes open is not that difficult, but doing it eyes closed seems to be difficult. Why?
Balance is tricky and depends on a lot of things, including, to some degree, your sight. Balance is achieved and maintained by a complex set of sensorimotor control systems that include sensory input from vision (sight), proprioception (touch), and the vestibular system (motion, equilibrium, spatial orientation); integration of that sensory input; and motor output to the eye and body muscles. Your vision helps you see where your head and body are in relationship to the world around you and to sense motion between you and your environment.
Very simply put, to blindfold someone is to take out the contribution of vision to balance. So, if the other systems involved in balance are off even a bit, blindfolding someone will interfere with their ability to keep their balance, even on two legs, let alone one.
Here are those systems in more detail.
Proprioception. In your joints, muscles, tendons, skin, and other areas, you have proprioceptors, that tell your brain where yoor aody parts (arms, legs, etc) are in relation to self in space. Because we live on a planet with gravity, even if we are blindfolded, if we are standing correctly, our feet will sent our brain signals that will in turn allow the brain to judge our position in space with regard to gravity and we will keep our feet firmly planted on the ground in a way which will maintain us upright in a constantly regulated position (if we sense less pressure from the bottom of our right great toe, we will adjust by leaning forward and to the right a bit until out=r brain says, "ok, right there!" This happens from all the skin and joint proprioceptors responding at the same time. Notice that once vision is blocked, our bodies become relatively very still; we focus on our balance. We move tentatively.
As people age, they lose some of their propioceptive abilities. That's one of the reasons for falls in the elderly.
Inner ear. In our inner ear are the semi-circular canals, three on each side, which are oriented at ~90° to each other so we cover 3 dimensions. In these canals is fluid which sloshes about when we move our heads, change directions, bend over, etc. The fluid moving about is detected by fine hair-like projections on the sensory cells lining the canal called stereocilia - like a wave would be detected by your leg in shallow ocean water. These semicircular canals help us to remain oriented in space. Now, the semicircular canals are also attached to otolith organs - two chambers which are concerned with gravity. They have fine crystals in them - called otoconia - which move according to our head's position with relationship to "down", or gravity.
If we stand perfectly still (probably not possible without support), everything is read by the vestibular system (the semicircular canals and the otolithic organs) as normal, and we stay upright (there is a constant communication between our vestibular system and our brain). So, when blindfolded, we can stay upright even if we shake our heads, bend them forward or back, etc (within reason). As we age, things can degenerate in this vestibular system, which affects our balance, and results in more falls in the elderly. (Diseases in the young can also affect this.)
Vestibulo-ocular Reflex. A complicated system which reflexly coordinated input from your vestibular system and your vision in head movement. If someone suddenly turns your head, your eyes will move in the equal and opposite manner, that is, your eyes will stay focused on the object on which they were focused just before your head was moved. This system is what allows you to easily keep reading something while you shake your head from side to side (not the case if you move the print side to side).
When things start going wrong (in a very slow manner) with your vestibular or proprioception systems, your eyes can help compensate by orienting you to the horizon, etc., in a manner, correcting for some 'off' calculations in your vestibular system. Vision helps us to identify orientation with respect to gravity, determine direction and speed of movement, etc., to a much lesser degree than the other systems described but enough to have an impact. So, in people whose vision is failing, either their other systems have to be in really good working order (e.g. why young blind people aren't always falling over) or they have problems with falling.
The Human Balance System (easy)
Researchers find diminished balance in those with poor vision (easy)
Visual influences on balance
Baloh and Honrubia's Clinical Neurophysiology of the Vestibular System By Robert William Baloh, Kevin Kerber, MD
Vestibular Occular Reflex